Properties

Label 5.5.70601.1-47.1-a2
Base field 5.5.70601.1
Conductor norm \( 47 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 5.5.70601.1

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 3 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{4}-2a^{3}-9a^{2}+3a+3\right){x}{y}+\left(a^{4}-6a^{2}-2a+4\right){y}={x}^{3}+\left(a^{4}-6a^{2}-a+3\right){x}^{2}+\left(47a^{4}-31a^{3}-246a^{2}+10a+150\right){x}-159a^{4}+108a^{3}+829a^{2}-51a-493\)
sage: E = EllipticCurve([K([3,3,-9,-2,2]),K([3,-1,-6,0,1]),K([4,-2,-6,0,1]),K([150,10,-246,-31,47]),K([-493,-51,829,108,-159])])
 
gp: E = ellinit([Polrev([3,3,-9,-2,2]),Polrev([3,-1,-6,0,1]),Polrev([4,-2,-6,0,1]),Polrev([150,10,-246,-31,47]),Polrev([-493,-51,829,108,-159])], K);
 
magma: E := EllipticCurve([K![3,3,-9,-2,2],K![3,-1,-6,0,1],K![4,-2,-6,0,1],K![150,10,-246,-31,47],K![-493,-51,829,108,-159]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-2a^3-3a^2+5a)\) = \((a^4-2a^3-3a^2+5a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 47 \) = \(47\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^4+2a^3+4a^2-8a+10)\) = \((a^4-2a^3-3a^2+5a)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -103823 \) = \(-47^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5215486471}{103823} a^{4} - \frac{9695772757}{103823} a^{3} - \frac{17753063912}{103823} a^{2} + \frac{25718276651}{103823} a - \frac{6096977969}{103823} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(a^{4} - a^{3} - 5 a^{2} + a + 3 : -2 a^{4} + a^{3} + 11 a^{2} + a - 7 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 827.77199692779710298409939764874693779 \)
Tamagawa product: \( 3 \)
Torsion order: \(2\)
Leading coefficient: \( 2.33650421 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((a^4-2a^3-3a^2+5a)\) \(47\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 47.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.